29. Letta, a, . . ., a, be fixed real numbers and define a functiona)x-aa2)...(x- anWhat is limf(x) ? For some a # a1, a2,, an, compute lim f(x).x→ax→ a

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Solution

Let the given function be

f( x )=( x− a 1 )( x− a 2 )( x− a 3 )…( x− a n )

We have to find the value of the given function at limit x→ a 1 .

Applying the concept of limits to the given function, we get

lim x→ a 1 = lim x→ a 1 ( x− a 1 )( x− a 2 )( x− a 3 )…( x− a n ) = lim x→ a 1 ( x− a 1 ) lim x→ a 1 ( x− a 2 ) lim x→ a 1 ( x− a 3 )… lim x→ a 1 ( x− a n ) =( a 1 − a 1 )( a 1 − a 2 )( a 1 − a 3 )…( a 1 − a n ) =0

Therefore, the value of the lim x→ a 1 f( x )=0 .

Now, we have to find the value of the same function at limit x→a .

Applying the concept of limits to the given function, we get

lim x→a = lim x→a ( x− a 1 )( x− a 2 )( x− a 3 )…( x− a n ) = lim x→a ( x− a 1 ) lim x→a ( x− a 2 ) lim x→a ( x− a 3 )… lim x→a ( x− a n ) =( a− a 1 )( a− a 2 )( a− a 3 )…( a− a n )

Thus, the value of the lim x→a f( x )=( a− a 1 )( a− a 2 )( a− a 3 )…( a− a n ) .

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